次のページを参考にしました。
Crate reqwest

プロジェクトの作成

cargo new http_post --bin

Cargo.toml

[package]
name = "http_post"
version = "0.1.0"
edition = "2018"

#

[dependencies]
reqwest = { version = "0.10", features = ["json"] }
tokio = { version = "0.2", features = ["full"] }
serde = { version = "1.0", features = ["derive"] }
serde_json = "1.0"

src/main.rs

// --------------------------------------------------------------------
/*
	http_post.rs

					Jul/15/2020
*/
// --------------------------------------------------------------------
use std::collections::HashMap;

# [tokio::main]

async fn main() -> Result<(), Box<dyn std::error::Error>> {
	eprintln!("*** 開始 ***");
	let mut map = HashMap::new();
	map.insert("user", "jiro");
	map.insert("password", "123456");

	let client = reqwest::Client::new();
	let url = "https://httpbin.org/post";
	let resp = client.post(url)
		.json(&map)
		.send()
		.await?;

//	println!("{:#?}", resp);

	let body = resp.text().await? ;
//	println!("{:#?}", body);

	let json: serde_json::Value = serde_json::from_str(&body)?;

	let obj = json.as_object().unwrap();

	for (key,value) in obj.iter() {
		println!("{}\t{}",key,value);
		}

	eprintln!("*** 終了 ***");
	Ok(())
}

// --------------------------------------------------------------------

コンパイル

cargo build

実行結果

$ cargo run
    Finished dev [unoptimized + debuginfo] target(s) in 0.06s
     Running `target/debug/http_post`
*** 開始 ***
args	{}
data	"{\"user\":\"jiro\",\"password\":\"123456\"}"
files	{}
form	{}
headers	{"Accept":"*/*","Content-Length":"35","Content-Type":"application/json","Host":"httpbin.org","X-Amzn-Trace-Id":"Root=1-5f0eb9e1-064b695a2db532c4d57d894e"}
json	{"password":"123456","user":"jiro"}
origin	"219.126.137.21"
url	"https://httpbin.org/post"
*** 終了 ***

次のバージョンで確認しました。

$ rustc --version
rustc 1.43.0

$ cargo --version
cargo 1.43.0